An Ascoli theorem for sequential spaces

Abstract

Abstract. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C * -algebras.2000 Mathematics Subject Classification. 54A20, 54C35, 54E15.1. Introduction. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. Such general theorems are inspired by the classical Ascoli theorem, proved by G. Ascoli (and independently by C. Arzelà) in the 19th century (see [3,4]). It characterizes compactness of sets of continuous real-valued functions on the interval [0, 1] with respect to the topology of uniform convergence. Since then, many related theorems have been proved, for example, characterizing compactness of sets of continuous functions from a topological to a uniform space (see [6]), of uniformly continuous functions from a merotopic to a uniform space (see [5]) of continuous functions between topological spaces (see [14,17]). As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natural function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct. Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categorical setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory. Wyler himself developed new examples of Ascoli theorems for sets of continuous functions between limit spaces and of uniformly continuous functions from a uniform convergence space to a pseudo-uniform space (see [24]).In this paper, we present another setting for an Ascoli theorem: we choose the construct L of sequential spaces. Sequential spaces were already introduced at the beginning of the century by Fréchet (see [9,10]) and Urysohn (see [2,23]), even before topological spaces were axiomatized. Since then they have extensively been used as a tool in topology and analysis. Since the 60s sequential structures have been investigated from a categorical point of view (for categorical background, we refer to [1]);

Pointwise convergence and Ascoli theorems for nearness spaces

“…In 1993, [5] studied a version of the Ascoli's theorem for set valued proximally continuous functions. In 2001, [17] proved a version of Ascoli's theorem for sequential spaces. As far as we know, no nearness space version of the Ascoli's theorem has been established yet at this time.…”